Sociology 5811: TTests for Difference in Means

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Sociology 5811T-Tests for Difference in Means

• Wes Longhofer, pinch-hitting for Evan Schofer

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Strategy for Mean Difference

• We never know true population means
• So, we never know true value of difference in
  means
• So, we dont know if groups really differ
• If we can figure out the sampling distribution of
  the difference in means
• We can guess the range in which it typically
  falls
• If it is improbable for the sampling distribution
  to overlap with zero, then the population means
  probably differ
• An extension of the Central Limit Theorem
  provides information necessary to do calculations!

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Sampling Distribution for Difference in Means

• The mean (Y-bar) is a variable that changes
  depending on the particular sample we took
• Similarly, the differences in means for two
  groups varies, depending on which two samples we
  chose
• The distribution of all possible estimates of the
  difference in means is a sampling distribution!
• The sampling distribution of differences in
  means
• It reflects the full range of possible estimates
  of the difference in means.

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Mean Differences for Small Samples

• Sample Size rule of thumb
• Total N (of both groups) gt 100 can safely be
  treated as large in most cases
• Total N (of both groups) lt 100 is possibly
  problematic
• Total N (of both groups) lt 60 is considered
  small in most cases
• If N is small, the sampling distribution of mean
  difference cannot be assumed to be normal
• Again, we turn to the T-distribution.

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Mean Differences for Small Samples

• To use T-tests for small samples, the following
  criteria must be met
• 1. Both samples are randomly drawn from normally
  distributed populations
• 2. Both samples have roughly the same variance
  (and thus same standard deviation)
• To the extent that these assumptions are
  violated, the T-test will become less accurate
• Check histogram to verify!
• But, in practice, T-tests are fairly robust.

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Mean Differences for Small Samples

• For small samples, the estimator of the Standard
  Error is derived from the variance of both groups
  (i.e. it is pooled)
• Formulas

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Probabilities for Mean Difference

• A T-value may be calculated

• Where (N1 N2 2) refers to the number of
  degrees of freedom
• Recall, t is a family of distributions
• Look up t-dist for N1 N2 -2 degrees of
  freedom.

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T-test for Mean Difference

• Back to the example 20 boys 20 girls
• Boys Y-bar 72.75, s 8.80
• Girls Y-bar 78.20, s 9.55
• Lets do a hypothesis test to see if the means
  differ
• Use a-level of .05
• H0 Means are the same (mboys mgirls)
• H1 Means differ (mboys ? mgirls).

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T-test for Mean Difference

• Calculate t-value

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T-Test for Mean Difference

• We need to calculate the Standard Error of the
  difference in means

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T-Test for Mean Difference

• We also need to calculate the Standard Error of
  the difference in means

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T-test for Mean Difference

• Plugging in Values

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T-test for Mean Difference

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T-Test for Mean Difference

• Question What is the critical value for a.05,
  two-tailed T-test, 38 degrees of freedom (df)?
• Answer Critical Value approx. 2.03
• Observed T-value 1.88
• Can we reject the null hypothesis (H0)?
• Answer No! Not quite!
• We reject when t gt critical value

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T-Test for Mean Difference

• The two-tailed test hypotheses were

• Question What hypotheses would we use for the
  one-tailed test?

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T-Test for Mean Difference

• Question What is the critical value for a.05,
  one-tailed T-test, 38 degrees of freedom (df)?
• Answer Around 1.684 (40 df)
• One-tailed test T 1.88 gt 1.684
• We can reject the null hypothesis!!!
• Moral of the story
• If you have strong directional suspicions ahead
  of time, use a one-tailed test. It increases
  your chances of rejecting H0.
• But, it wouldnt have made a difference at a.01

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Another Example

• Question Do the mean batting averages for
  American League and National League teams differ?
• Use a random sample of teams over time
• American League Y-bar .2677, s .0068, N14
• National League Y-bar .2615, s .0063, N16
• Lets do a hypothesis test to see if the means
  differ
• Use a-level of .05
• H0 Means are the same (mAmerican mNational)
• H1 Means differ (mAmerican ? mNational)

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T-test for Mean Difference

• Calculate t-value

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T-Test for Mean Difference

• We need to calculate the Standard Error of the
  difference in means

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T-Test for Mean Difference

• We also need to calculate the Standard Error of
  the difference in means

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T-test for Mean Difference

• Plugging in Values

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T-test for Mean Difference

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T-Test for Mean Difference

• Question What is the critical value for a.05,
  two-tailed T-test, 28 degrees of freedom (df)?
• Answer Critical Value approx. 2.05
• Observed T-value 2.58
• Can we reject the null hypothesis (H0)?
• Answer Yes
• We reject when t gt critical value
• What if we used an a-level of .01?
• Critical value2.76

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T-Test for Mean Difference

• Question What if you wanted to compare 3 or
  more groups, instead of just two?
• Example Test scores for students in different
  educational tracks honors, regular, remedial
• Can you use T-tests for 3 groups?
• Answer Sort of You can do a T-test for every
  combination of groups
• e.g., honors reg, honors remedial, reg
  remedial
• But, the possibility of a Type I error
  proliferates 5 for each test
• With 5 groups, chance of error reaches 50
• Solution ANOVA.